Seminars and Colloquia by Series

Obstructions to embeddings in 4-manifolds

Series
Geometry Topology Seminar
Time
Friday, April 9, 2021 - 13:00 for 1 hour (actually 50 minutes)
Location
On line
Speaker
Anubhav MukherjeeGeorgia Tech

Please Note: Note special day and time.

In this talk I will discuss some new properties of an invariant for 4-manifold with boundary which was originally defined by Nobuo Iida. As one of the applications of this new invariant I will demonstrate how one can obstruct a knot from being h-slice (i.e bound a homologically trivial disk)  in 4-manifolds. Also, this invariant can be useful to detect exotic smooth structures of 4-manifolds. This a joint work with Nobuo Iida and Masaki Taniguchi.

Right-veering open books and the Upsilon invariant

Series
Geometry Topology Seminar
Time
Monday, April 5, 2021 - 14:00 for 1 hour (actually 50 minutes)
Location
ONLINE
Speaker
Diana HubbardBrooklyn College, CUNY

Fibered knots in a three-manifold Y can be thought of as the binding of an open book decomposition for Y. A basic question to ask is how properties of the open book decomposition relate to properties of the corresponding knot. In this talk I will describe joint work with Dongtai He and Linh Truong that explores this: specifically, we can give a sufficient condition for the monodromy of an open book decomposition of a fibered knot to be right-veering from the concordance invariant Upsilon.  I will discuss some applications of this work, including an application to the Slice-Ribbon conjecture.

Topology of the Shift Locus via Big Mapping Class Groups by Yan Mary He

Series
Geometry Topology Seminar
Time
Monday, March 29, 2021 - 14:00 for 1 hour (actually 50 minutes)
Location
ONLINE
Speaker
Yan Mary HeUniversity of Toronto

The shift locus of (monic and centered) complex polynomials of degree d > 1 is the set of polynomials whose filled-in Julia set contains no critical points. Traversing a loop in the shift locus gives rise to a holomorphic motion of Cantor Julia sets, which can be extended to a homeomorphism of the plane minus a Cantor set up to isotopy. Therefore there is a well-defined monodromy representation from the fundamental group of the shift locus to the mapping class group of the plane minus a Cantor set. In this talk, I will discuss the image and the kernel of this map as well as a combinatorial model for the shift locus. This is joint work with J. Bavard, D. Calegari, S. Koch and A. Walker.

Infinite-type surfaces and the omnipresent arcs by Tyrone Ghaswala

Series
Geometry Topology Seminar
Time
Monday, March 22, 2021 - 14:00 for 1 hour (actually 50 minutes)
Location
ONLINE
Speaker
Tyrone GhaswalaCIRGET, Université du Québec à Montréal

Please Note: A pre-talk will be given at 1 and office hours will be held at 3 (following the seminar talk).

In the world of finite-type surfaces, one of the key tools to studying the mapping class group is to study its action on the curve graph. The curve graph is a combinatorial object intrinsic to the surface, and its appeal lies in the fact that it is infinite-diameter and $\delta$-hyperbolic. For infinite-type surfaces, the curve graph disappointingly has diameter 2. However, all hope is not lost! In this talk I will introduce the omnipresent arc graph and we will see that for a large collection of infinite-type surfaces, the graph is infinite-diameter and $\delta$-hyperbolic. The talk will feature a new characterization of infinite-type surfaces, which provided the impetus for this project.

This is joint work with Federica Fanoni and Alan McLeay

Big mapping class groups and rigidity of the simple circle by Lvzhou Chen

Series
Geometry Topology Seminar
Time
Monday, March 15, 2021 - 14:00 for 1 hour (actually 50 minutes)
Location
ONLINE
Speaker
Lvzhou ChenUT Austin

Please Note: Office hours will be held 3-4 pm.

Surfaces of infinite type, such as the plane minus a Cantor set, occur naturally in dynamics. However, their mapping class groups are much less studied and understood compared to the mapping class groups of surfaces of finite type. Many fundamental questions remain open. We will discuss the mapping class group G of the plane minus a Cantor set, and show that any nontrivial G-action on the circle is semi-conjugate to its action on the so-called simple circle. Along the way, we will discuss some structural results of G to address the following questions: What are some interesting subgroups of G? Is G generated by torsion elements? This is joint work with Danny Calegari.

Julia sets with Ahlfors-regular conformal dimension one by InSung Park

Series
Geometry Topology Seminar
Time
Monday, February 22, 2021 - 14:00 for 1 hour (actually 50 minutes)
Location
ONLINE
Speaker
InSung ParkIndiana University Bloomington

Please Note: Office hours will be held 3-4pm EST.

Complex dynamics is the study of dynamical systems defined by iterating rational maps on the Riemann sphere. For a rational map f, the Julia set Jf  is a beautiful fractal defined as the repeller of the dynamics of f. Fractal invariants of Julia sets, such as Hausdorff dimensions, have information about the complexity of the dynamics of rational maps. Ahlfors-regular conformal dimension, abbreviated by ARconfdim, is the infimum of the Hausdorff dimension in a quasi-symmetric class of Ahlfors-regular metric spaces. The ARconfdim is an important quantity especially in geometric group theory because a natural metric, called a visual metric, on the boundary of any Gromov hyperbolic group is determined up to quasi-symmetry. In the spirit of Sullivan's dictionary, we can use ARconfdim to understand the dynamics of rational maps as well. In this talk, we show that the Julia set of a post-critically finite hyperbolic rational map f has ARconfdim 1 if and only if there is an f-invariant graph G containing the post-critical set such that the dynamics restricted to G has topological entropy zero.  

Braids, quasimorphisms, and slice-Bennequin inequalities.

Series
Geometry Topology Seminar
Time
Monday, February 8, 2021 - 14:00 for 1 hour (actually 50 minutes)
Location
ONLINE
Speaker
Peter FellerETH Zurich

The writhe of a braid (=#pos crossing - #neg crossings) and the fractional Dehn twist coefficient of a braid (a rational number that measures "how much the braid twists") are the two most prominent examples of what is known as a quasimorphism (a map that fails to be a group homomorphism by at most a bounded amount) from Artin's braid group on n-strands to the reals.
We consider characterizing properties for such quasimorphisms and talk about relations to the study of knot concordance. For the latter, we consider inequalities for quasimorphism modelled after the so-called slice-Bennequin inequality:
writhe(B) ≤ 2g_4(K) - 1 + n for all n-stranded braids B with closure a knot K.
Based on work in progress.

Symmetric knots and the equivariant 4-ball genus

Series
Geometry Topology Seminar
Time
Monday, February 1, 2021 - 14:00 for 1 hour (actually 50 minutes)
Location
ONLINE
Speaker
Ahmad IssaUniversity of British Columbia

Given a knot K in the 3-sphere, the 4-genus of K is the minimal genus of an orientable surface embedded in the 4-ball with boundary K. If the knot K has a symmetry (e.g. K is periodic or strongly invertible), one can define the equivariant 4-genus by only minimising the genus over those surfaces in the 4-ball which respect the symmetry of the knot. I'll discuss some work with Keegan Boyle trying to understanding the equivariant 4-genus.

The asymptotic dimension of big mapping class groups

Series
Geometry Topology Seminar
Time
Monday, January 25, 2021 - 14:00 for 1 hour (actually 50 minutes)
Location
ONLINE
Speaker
Yvon VerberneGeorgia Institute of Technology

Please Note: Dan Margalit is inviting you to a scheduled Zoom meeting. https://zoom.us/j/94410378648?pwd=TVV6UDd0SnU3SnAveHA1NWxYcmlTdz09 Meeting ID: 944 1037 8648 Passcode: gojackets

In 2010, Bestvina-Bromberg-Fujiwara proved that the mapping class group of a finite type surface has finite asymptotic dimension. In contrast, we will show the mapping class group of an infinite-type surface has infinite asymptotic dimension if it contains an essential shift. This work is joint with Curtis Grant and Kasra Rafi.

Identifying Dehn Functions of Bestvina--Brady Groups From Their Defining Graphs

Series
Geometry Topology Seminar
Time
Monday, January 11, 2021 - 14:00 for 1 hour (actually 50 minutes)
Location
ONLINE
Speaker
Yu-Chan ChangEmory University

Please Note: https://zoom.us/j/8833025617?pwd=R1FvQWp1MVlRSTVBdFZNejE3ZURmUT09 Meeting ID: 883 302 5617

Bestvina--Brady groups are subgroups of right-angled Artin groups, and their Dehn functions are bounded above by quartic functions. There are examples of Bestvina--Brady groups whose Dehn functions are linear, quadratic, cubic, and quartic. In this talk, I will give a class of Bestvina--Brady groups that have polynomial Dehn functions, and we can identify the Dehn functions by the defining graphs of those Bestvina--Brady groups. 

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